Uniform forcing and immune sets in graphs and hypergraphs

TL;DR

This paper extends the concept of zero forcing from graphs to hypergraphs, analyzing minimal forcing and immune sets, and providing characterizations for complete hypergraphs and uniform clutters.

Contribution

It introduces two generalizations of zero forcing to hypergraphs and characterizes minimal forcing and immune sets in these contexts, including for complete hypergraphs.

Findings

Characterization of minimal forcing and immune sets in hypergraphs

Formulation of immune sets using neighborhoods

Full characterization of forcing and immune uniform clutters

Abstract

Zero forcing is an iterative coloring process on a graph that has been widely used in such different areas as the modelling of propagation phenomena in networks and the study of minimum rank problems in matrices and graphs. This paper deals with zero forcing on hypergraphs. (Representing a network by a hypergraph allows us to account for its community structure and for more general interactions between different subsets of nodes.) We consider two natural generalizations to hypergraphs of zero forcing on graphs (one of them already known) and, for each one of these generalizations we look into two clutters that play a significant role in the forcing process: the clutter of minimal forcing sets and the one of minimal immune sets. A formulation of immune sets in terms of neighbourhoods (hence without making reference to the iterative zero forcing process) is presented, highlighting the different behaviour of the distinct forcing rules. Moreover, we obtain the families of minimal forcing and minimal immune sets in the case of complete hypergraphs and we provide a full characterization of forcing and immune uniform clutters, both in the graph and in the hypergraph case.